IMP Alice Day 8 - Having Your Cake and Drinking Too

When I saw that the teacher's guide allowed 60 minutes for this activity I thought there was no way. I thought we would get through it quicker...haha! We didn't even finish and we had OVER an hour. This activity helps the students to explore and HOPEFULLY discover the rule for dividing powers with the same base. I love it! It is really revealing the lack of number sense that my students have. The questions are not really that difficult...that is why I thought we would breeze right through.

I have allowed myself to "sit" on this activity and not rush it because I really believe it might help to cement the concept. I am going to be absent tomorrow from my classes and I have some good exploration worksheets on exponents to leave for my students. However, it has reminded me how blessed we are to have found a book that combines the exploration with the context. Someone had a wonderful imagination!

**Update 3/7/15** Original post was made 3/16/15
This year when I taught this lesson I took the time to help the students to explore #2. I did not let anyone blurt out an answer when we first started looking at it. I have been using the random integer generator on the TI-83 (which is awesome but I just now figured out how to do it!!) to call on students. In my classes I just happened to call on students who were unsure of what to do so I just asked them to start by making a guess. I told them to tell me a number of ounces of cake and a number of ounces of beverage and then I showed them how to "test" their answer to see if this would give an answer where Alice's height was multiplied by 8. Then I randomly called on more students. The first 3 students guessed more beverage than cake (ugh!) so I asked the next student to make a "conjecture" on whether or not more beverage than cake would EVER allow Alice's height to be multiplied by 8. My 2nd class arrived at the correct number of ounces much faster than my 1st one did but I feel that modeling to the students how to "guess and check" was valuable. I am always telling them not to ever erase their guesses. I want them to learn to look back over the ideas that didn't work in order to help them to identify new theories that might work!! Anyway...the day I taught this lesson this year I felt really good about what we had accomplished:)

IMP Day 43 - Making "formal" connections

I have thoroughly enjoyed the Moving Along activity and decided to "sit down" here and make some formal connections to the slope formula and slope-intercept form. However, we first did the activity without using either formulas and it was AMAZING! We actually have been using slope-intercept form but they just didn't know it as y=mx+b. The IMP book uses y=ax+b and calls b the "starting point" and a the "rate of change."

Today I gave the class a "warm-up" where they were asked to find the slope of a line and write the equation of the line given 2 points. There was no context given and they did great! There were no formulas on the board but most of the students proceeded to put the 2 points in a table and then fill in the x-values from 0 to the highest x-value given. They go back to 0 for x because we have drilled the fact that x-coordinate of the starting point is always 0. We have also tried to drill that the starting point is always on the y-axis but they seem to forget that sometimes...

After the warm-up, which most of them got with a table, I had them add the slope formula and slope-intercept formula to their notes. I told them that they would receive a reference page on their end-of-course (EOC) algebra exam and that I wanted them to be familiar with the formulas. We went over how we could have used the formulas for the warm-up and then I gave them another problem.

I used this weird effect on my picture to make it a little more readable. 

He writes so light I know it is a little difficult to read. However, Raul (whose paper is above) was the first student finished finding the slope and equation of the line! Then I gave the class a problem where the slope was a fraction thinking that they would resort to using the formula (and most of them did). However, Raul and one other student STILL used the table to get the equation. Their "number sense" is very good and thinking in fractions (or decimals) did not bother those 2 a bit! We have only been writing equations of lines given 2 points for 2 days and there are many more students who do it correctly than I have seen in the past.

It is exciting to realize that there are students who really benefit from their exposure to the different methods that can be used to write equations. I guess that since I am so accustomed to using the formulas I thought they would automatically start using them. However, the majority of my students are still using tables! I don't think I have ever used a table to find the slope or y-intercept so I am getting an education too!

Math teachers are "Formula Babies" - we need to be more natural!!

My teaching buddy, Sonya New, and I are writing this post together! We have learned this week that we are FORMULA BABIES!! We ran across some problems where students need to write equations given 2 points and thought the students would just HAVE to have slope and point-slope formulas because that is how we learned to do it ourselves. BAHAHAHAHAHAHA!!

What an education we have received! I was so concerned about how they would find the "starting point" or y-intercept without formulas! Earlier this week Sonya's honors algebra realized that once they found the rate of change (slope) they could multiply the x coordinate by the slope and find the b (or starting point) by figuring out what to add or subtract to get y. I know that makes no sense when you read it! However, it took me and Sonya 2 WHOLE DAYS to realize that what they are doing is using the slope-intercept formula to solve for b. We felt STUPID!

ALSO, I had students to use tables to find rate of change and then extend the table "back" to zero to find the starting point or y-intercept. I had one student who hated the formulas yet got EVERY SINGLE PROBLEM correct using tables. Even problems with fractional slopes!! IT WAS AMAZING AND EYE-OPENING!!

Then we laughed about the fact that we are "formula babies" and have come to the conclusion that we need to understand that the logical (or NATURAL instead of FORMULA) way to write equations makes more sense to our students.

P.S. - Sonya is a new mother and I have had 3 breast-fed babies myself...so we couldn't resist the analogy.

Day 22 IMP - Ox Expressions and Ox Expressions at Home

Today I started class by having each group compare to see who came up with the most meaningful algebraic expressions using the Ox Expressions chart. It was a sad comparison because the vast majority of my students did not do their homework. I had told them that the winner would receive bonus points to try to motivate them.

After they had time to compare, I took up each of the "winning" papers from each group and gave a piece of candy to each of those students (candy always motivates!). Then I put their papers under the document camera and we assessed each expression to see if it was meaningful. This was a tedious process but I am not sure how else to get them to go through and determine meaningful expressions! Unit analysis really helps! Again...we did unit analysis before we started the Meaningful Math books. I do not know if they REALLY understood it then and I am not sure how many students REALLY understand it now, but for the few that do it seems to help to verify whether or not expressions have useful meaning. Also...I talked about "like terms." For example, it means nothing to add the number of wagons plus the number of gallons of water a person drinks per day. I point out that adding wagons to gallons doesn't mean much. However, if you add the number of men plus the number of women plus the number of children it is MEANINGFUL because they are all the same unit - people!

After we trudged through determining whether or not the expressions were meaningful and found a winner for each class (one girl had 14!), I made them do the Ox Expressions at Home activity as a quiz. I look forward to seeing how well they did. These concepts seem to be difficult to "drive home."

Lastly, Dr. Montgomery, our instructional partner and former human anatomy and physiology teacher, came in and discussed the diseases in "If I Could See This Thing." She also has created death certificates and tombstones for me and Mrs. New to use with our classes. I have to prepare for tomorrow's lesson but I think a certain percentage of the people in our wagon trains have to die due to these diseases. More tomorrow...

IMP Day 5 - Calculator Exploration...and The Broken Eggs

On Day 5 I chose to go ahead and do the Calculator Exploration with my classes. I was very tempted to skip it because we are going to have to trim some items in the books since we are starting 7 weeks late. However, it was a Friday and I thought the students might enjoy it. What is funny is they taught me things about the TI-83 calculators that I didn't know. I have never played with any of the Apps and of course this generation is all about Apps. I also learned how to dim and brighten the screen AND how to convert decimals to fractions and vice versa. I asked at the end of class how many students were more excited about our new books now that they realize that they will get to use the graphing calculators and about 5 or 6 raised their hands. That is improvement. Right now many of the students are not happy with our new books because they are more wordy and they are required to explain and describe their reasoning more. The way the curriculum is written we will EASILY be meeting our new literacy standards. I am excited about the amount of reflection and explaining that the book asks for because I was interested in having my students journal anyway. I just have a hard time implementing journals in my classroom.  After the calculator exploration I gave my students time to start writing their POW write up for The Broken Eggs.

Thursday I had a student in my 5th period class tell me that he had come up with a way to find an endless number of solutions to The Broken Eggs. He told me verbally but he lost me! I told him to write it down and put it on my desk (the bell was about to ring). He had written down to take the previous answer, multiply it by 2, and then add 119. I used 301 as the "previous answer" and used his method and it worked. I went and showed Sonya New to see if she could help me figure out how he came up with it. We were stumped! When he came to class on Friday I asked him to explain. He said that he found 301 by adding 7 over and over again. The class had already discussed that the answer had to be an odd multiple of 7. This young man had also decided that the number had to end with a 1 because to have 1 left over when putting the eggs in packages of 5 the number had to end in a 6 or a 1 and since the number had to be odd it had to end with a 1. So...he kept adding 7 until he came to the next multiple of 7 that ended in a 1 and it was 721. Then he said he was just playing with the numbers and decided to multiply 301 by 2 to see how close it was and it was 119 away. Then he decided to try to double 721 and add 119 to see if it worked to generate another solution and it did. WOW! That seems so random to me. I am still amazed.

A 4th grader blew my mind

My daughter was telling me today about how she figured out how to multiply 8 times 8 when she had her timed test in math. She is working on memorizing her multiplication facts for 2s, 5s, 9s, and squares. She has most of them down now but a few of them were still escaping her. Therefore she was telling me that she skipped 8x8 during her timed test and did it last. She said she figured it out by counting 8 5s on her fingers which comes up to 40. Then she took her 8 fingers she had up and added 3 more for each finger. She counted 41, 42, 43 for her first finger; then she added 44, 45, 46, for her second finger and then so on. She came up with 64 and I was amazed. It took me a minute to figure out what she had done. THE DISTRIBUTIVE PROPERTY...8x8 = 8(5+3). She multiplied the 8x5 then did the 8x3 (by counting on her fingers...lol).  I asked her who showed her that and she said she just came up with it. I BEG TO DIFFER. She has had some great teachers and she is doing the dreaded GO MATH.

I keep hearing and seeing so many complaints about common core math. However, this little problem solver is a product of common core math. It is so hard to explain sometimes but I try. Common core teaches problem solving skills- not just rote memory. That does not mean that students won't memorize multiplication facts. They certainly will still memorize facts. However, they also use estimation and problem solving strategies that they can put into practice when they are not sure they are remembering their math facts correctly. I believe I have seen and heard phrases like friendly numbers and compatible numbers. I see posts about the "new way" to solve math problems and how long they take. Please consider that the students are learning problem solving strategies that will go with them as the problems get too hard to know by memory. The simpler problems are used to illustrate the concept but the concept applies as the problems get more difficult. It is similar to making students show their work on 1-step equations even though they can do them in their heads. We are teaching them the PROCESS so that when the equations get more difficult and involve more steps they can understand the correct way to solve them.

I like common core math. I love the focus on the problem solving skills and the math practice standards. I also love the incredible amount of resources that can be found to help teach algebra. I do think that schools need to have "tutoring" for parents so that they can understand the reasons for the different strategies that are being taught for solving their math problems. Faster is not always better. Comprehension and retention is the key!